We started today by recapping the Crow & the Pitcher activity from yesterday. I started with a glass that was more than half full of water and asked what we needed to do first. They said that we needed to measure the height of the water so I handed over the glass and a ruler to a student. He measured by placing the ruler inside the glass. I asked the rest of the class if it was okay to do this. One student said no because the "0 cm" of the ruler is not actually at the end of the ruler. Another said that by placing the ruler inside the glass, the height of the water would increase. I thanked the student who was measuring for having done that as it led to a great discussion.
Here is the 2nd attempt at measuring:
At first he said "...about 12.2 cm", to which I replied "About? No. We want an exact measurement!". He tried again and here are the results:
I asked my students to use their work from yesterday to determine the number of marbles needed to get the water to the very top of the glass. I asked them to write their answers on the board (without names) and this is what they got:
Quite the range! I collected some data myself and did my own calculations which I went over with them.
From the work they did yesterday, they knew the relationship was linear. Here is my data plotted with Desmos:
I made sure they understood all of the parts of the equation they were using by having them explain each element to the class.
We then talked about which numbers went where and why before calculating the number of marbles my model predicted.
Here it is with Desmos:
Notice that their regression line found the same number of marbles that I had predicted. And then for the actual test... Here is the glass with 94 marbles. So close!
I continued adding marbles until the water was just about to drip over the edge.
My model's prediction was closer than any of my students'. I assume this had a lot to do with their precision when measuring yesterday. It gave me a good opportunity to talk about how to do a good job when collecting data.
I really like this activity. It definitely got my students thinking and talking about math, and working together. Clearly, you need a lot of marbles! I will have to buy more for next time. I would also recommend getting marbles that are all the same. None of my students got through it quickly enough to try with the larger marbles I have, but perhaps another time. You could then extend this activity to linear systems by asking when two glasses would be at the same height with the same number of marbles. You would have to carefully choose the marbles and the starting heights, but this could work.
Next up, visual patterns! I had made this handout which they jumped right into. I wanted them to show me how the patterns were growing, and they did this pretty well. I did not see a lot of tables of values popping up which was really good. We took up the first two in class:
It was interesting to see how difficult some students found it to explain what their numbers meant. They seemed to enjoy these and will continue with more tomorrow.